Homework Statement
Why does Thomson scattering occur in the early universe?
Homework Equations
$$ e^{-} + \gamma \rightarrow e^{-} + \gamma $$
is a Thomson scattering process if:
$$ E_{\gamma} << m_{e}c^{2}$$
(Electrons are essenitally stationary)
The Attempt at a Solution
[/B]
Very...
Homework Statement
Let ## \mathbf{r} ## be the position of a point in a rigid body relative to some origin ##O##. Let ##\mathbf{R}## be the position of the centre of mass from that origin. ##\mathbf{r^{*}} = (\mathbf{r}-\mathbf{R})##. ## d\boldsymbol{\phi} ## is the infitesimal vector directed...
Sure, sure - the units are important and I entirely agree that ## p^{0}c ## has the correct units of energy. What I find confusing is not the units, but the idea that somehow a translation in time is related to my classical picture of energy :)
Good question - I presume it's the four velocity of the observer (these are directly from my lecture notes).
I guess another part of my confusion is the fact that we have :
$$
p^{\mu} = m \frac{dx^{\mu}}{d\tau}
\\[2mm]
p^{\mu} = \bigg(E,\vec{\dot{x}} \bigg) \\[2mm]
\implies...
Homework Statement
$$ E = -\vec{v_{obs}} \cdot \vec{p} $$
Where ## \vec{p} ## is the four momentum, and ## \vec{v_{obs}}## the velocity of the observer.
Homework Equations
The Attempt at a Solution
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This was a stated result in a GR course. I look through my SR notes and find that I...
It's not generally true that the off diagonal elements of the metric tensor are zero - we require that the tensor be symmetric, but that's not *necessarily* the same as it being diagonal? Even if I've got that wrong though...If you think of ##S^{2}##, then the metric definitely isn't just...
Homework Statement
Prove that, given a metric ##g_{ij}## such that ##ds^{2}=g_{ij}dx^{i}dx^{j}##, where ##x^{r} = x^{r}(\lambda)## , we have the following result for the arc length:
$$ L(p,q) = \int_{p}^{q} ds = \sqrt{ g_{ij} \frac{dx^{i}}{d \lambda} \frac{ dx^{j}}{d \lambda} } d \lambda $$...
Homework Statement
We have a 1-D lattice [a line] of ##L## sites. Sites are occupied with probability ##p##. Find the probability that a given site is a member of a cluster of size ##s##. (A cluster is a set of adjacent occupied sites. The cluster size is the number of occupied sites in the...
Ah, that's embarrassing. Out of interest - why is the cosine rule idea not valid? Please see picture.
**EDIT**: Oops, in the diagram ##\vec{A}## and ##\vec{C}## have the opposite phase to as written in the question.
Homework Statement
I'm looking at an E&M textbook - "Time-Harmonic Electromagnetic Fields". They state:
"A more general ##x ## polarized field is one consisting of waves traveling in opposite directions with unequal amplitudes - i.e :
(1) $$ E_{x} = Ae^{-jkz} +Ce^{jkz}$$
Let ## A ## and...
Oh, of course, because the diagonal of the square is ## \sqrt{2} ##. I see! Thanks. I think I am leaving ## z ## in the ##z## column though, but the vector field ## \vec{F} ## is ##\vec{F}=(-y,x,0) ## - or did you mean something else?
But am I correct in thinking now that the answer I first...
Ah, sorry. I forgot to say that the question stated ## a > \sqrt{2} ## , ## b >\sqrt{2} ## and ## a \neq b ##. Might this change things? (Though I admit, I don't know how they got these conditions).
Isn't the region of integration a square? ( I thought the definition of the region as ## 0 \leq x \leq 1 ## and ## 0 \leq y \leq 1 ## implied this?). Hence, no need for writing ## y=y(x)## in the limits...
and yes, I know I can get it from (2), but the question was phrased in a way that it...
Homework Statement
Let ## E ## be the ellipsoid:
$$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+z^{2}=1 $$
Let ## S ## be the part of the surface of ## E ## defined by:
$$ 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z > 0 $$
Let F be the vector field defined by $$ F=(-y,x,0)$$
A) Explain why ##...
Ah, no that's not at all what I meant. Sorry. I must have said something confusing :P
When I first wrote the above down, I was imagining the density as just a constant. Looking back I see no problem if ## \rho = \rho(x,y,z)## since ## dm## is infinitesimal. (though $\rho$ must be independent...
The component of the fluid velocity flowing through the small surface element ## dA ##. ## \rho ## may be a function of position and time, but why would this make the above invalid? Since we're using an infitesimal mass, ## \rho ## might (almost) as well be a constant, no? i.e. we're...
Homework Statement
Derive a mathematical relationship which encapsulates the principle of continuity in fluid flow.
Homework Equations
The Attempt at a Solution
Imagine we have a mass of fluid ## M##, of volume ##V##, bounded by a surface ##S##. If we take a small element of this volume...
Homework Statement
Don't understand why the inverse jacobian has the form that it does.
Homework Equations
$$ J = \begin{pmatrix} \frac{\partial{x}}{\partial{u}} & \frac{\partial{y}}{\partial{u}} \\ \frac{\partial{x}}{\partial{v}} & \frac{\partial{y}}{\partial{v}} \end{pmatrix} $$
$$...
Homework Statement
Evaluate:
I(y)= \int^{\frac{\pi}{2}}_{0} \frac{1}{y+cos(x)} \ dx if y > 1
Homework Equations
The Attempt at a Solution
I've never seen an integral like this before. I can see it has the form:
\int^{a}_{b} f(x,y) dx
I clearly can't treat it as one half of an exact...